Electronic Research Announcements in Mathematical Sciences (ERA-MS)

Linear approximate groups

Pages: 57 - 67, Volume 17, 2010      doi:10.3934/era.2010.17.57

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Emmanuel Breuillard - Laboratoire de Mathématiques, Bâtiment 425, Université Paris Sud 11, 91405 Orsay, France (email)
Ben Green - Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom (email)
Terence Tao - Department of Mathematics, UCLA, 405 Hilgard Ave, Los Angeles, CA 90095, United States (email)

Abstract: This is an informal announcement of results to be described and proved in detail in [3]. We give various results on the structure of approximate subgroups in linear groups such as $\SL_n(k)$. For example, generalizing a result of Helfgott (who handled the cases $n = 2$ and $3$), we show that any approximate subgroup of $\SL_n(\F_q)$ which generates the group must be either very small or else nearly all of $\SL_n(\F_q)$. The argument is valid for all Chevalley groups $G(\F_q)$. Extending work of Bourgain-Gamburd we also announce some applications to expanders, which will be proven in detail in [4].

Keywords:  Approximate groups, growth, expander graphs.
Mathematics Subject Classification:  20G40, 20N99.

Received: January 2010;      Available Online: September 2010.